Statistical Mechanics of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>Liquid Mixtures

The partition function method proposed by Feynman for pure liquid ${\mathrm{He}}^{4}$ and previously extended to treat pure liquid ${\mathrm{He}}^{3}$ by Kikuchi is here applied to liquid mixtures of ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$. The variation of the $\ensuremath{\lambda}$ point with mole fraction ${\mathrm{He}}^{3}$, the isotopic phase separation curve, and the excess functions of mixing are discussed. The theoretical $\ensuremath{\lambda}$ line is interpreted as a cooperative boson transition and follows the experimental results closely up to ${X}_{3}=0.5$ irrespective of the effective mass of the ${\mathrm{He}}^{3}$ atoms while above ${X}_{3}=0.5$ the $\ensuremath{\lambda}$ temperatures are too high. An asymmetric isotopic phase separation is found in the mixtures at temperatures below a critical temperature that depends slightly on further assumptions in the model but which is of the correct order of magnitude (1\ifmmode^\circ\else\textdegree\fi{}K). The phase separation is due to the quantum dynamical effects as opposed to the purely statistical effects arising out of the different inherent symmetries of the wave functions for ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$. The calculated excess Gibbs free energies of mixing become positive in "time" to effect the phase separation but are less positive than the experimental values and are in fact of the wrong sign above 1\ifmmode^\circ\else\textdegree\fi{}K. The calculated excess entropies of mixing are much too positive. The model used assumes zero excess volumes of mixing.

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